A second order SAP algorithm for risk and reliability based design optimization

This paper presents a decoupling approach for the efficient solution of Risk Optimization (RO) and Reliability Based Design Optimization (RBDO) problems. The proposed approach, SAP2nd, is a Sequential Approximate Programming (SAP) technique including second order terms obtained with the BFGS (Broyden-Fletcher-Goldfarb-Shanno) approximation for the Hessian. A first advantage of SAP2nd is that any reliability analysis method can be employed for the evaluation of the probabilities of failure and its sensitivities. Here, Polynomial Chaos Expansion (PCE) is employed for this purpose. Several benchmark problems are solved to study the efficiency, robustness and accuracy of SAP2nd. It is demonstrated that the inclusion of second order terms leads to: (i) a much more stable algorithm in comparison to a first order SAP algorithm, i.e. it was able to avoid convergence issues arising from cycling, and (ii) a more efficient algorithm since SAP2nd reduced the computational effort, in both RO and RBDO problems, when compared to the coupled PCE algorithm previously proposed by the authors. The use of PCE for the evaluation of the probabilities of failure and its sensitivities allowed SAP2nd to achieve much more accurate results when compared to FORM based approaches, requiring the same order of computational effort. Finally, SAP2nd using PCE for reliability and sensitivity analysis is well suited for RO and RBDO problems where the drawbacks of FORM based approaches prevail, especially cases with highly nonlinear limit state function and non Gaussian random variables.